What is the 'Black Scholes Model'
The Black Scholes model, also known as the BlackScholesMerton model, is a model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes the price of heavily traded assets follows a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price, and the time to the option's expiry.
BREAKING DOWN 'Black Scholes Model'
The Black Scholes model is one of the most important concepts in modern financial theory. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used now. It is regarded as one of the best ways of determining fair prices of options. The Black Scholes model requires five input variables: the strike price of an option, the current stock price, the time to expiration, the riskfree rate, and the volatility. Additionally, the model assumes stock prices follow a lognormal distribution because asset prices cannot be negative. Moreover, the model assumes there are no transaction costs or taxes; the riskfree interest rate is constant for all maturities; short selling of securities with use of proceeds is permitted; and there are no riskless arbitrage opportunities.
BlackScholes Formula
The Black Scholes call option formula is calculated by multiplying the stock price by the cumulative standard normal probability distribution function. Thereafter, the net present value (NPV) of the strike price multiplied by the cumulative standard normal distribution is subtracted from the resulting value of the previous calculation. In mathematical notation, C = S*N(d1)  Ke^(r*T)*N(d2). Conversely, the value of a put option could be calculated using the formula: P = Ke^(r*T)*N(d2)  S*N(d1). In both formulas, S is the stock price, K is the strike price, r is the riskfree interest rate and T is the time to maturity. The formula for d1 is: (ln(S/K) + (r + (annualized volatility)^2 / 2)*T) / (annualized volatility * (T^(0.5))). The formula for d2 is: d1  (annualized volatility)*(T^(0.5)).
Limitations of the Black Scholes Model
As stated previously, the Black Scholes model is only used to price European options and does not take into account that U.S. options could be exercised before the expiration date. Moreover, the model assumes dividends and riskfree rates are constant, but this may not be true in reality. The model also assumes volatility remains constant over the option's life, which is not the case because volatility fluctuates with the level of supply and demand.

Local Volatility
Local volatility is a volatility measure used in quantitative ... 
Merton Model
The Merton model is an analysis tool used to evaluate the credit ... 
Robert C. Merton
Robert C. Merton is a Nobel Prizewinning economist renowned ... 
Heston Model
The Heston Model is a type of stochastic volatility model used ... 
Trinomial Option Pricing Model
The trinomial option pricing model is an option pricing model ... 
Black Box Model
A black box model is a system using inputs and outputs to create ...

Investing
The Volatility Surface Explained
Learn about stock options and the "volatility surface," and discover why it is an important concept in stock options pricing and trading. 
Trading
Dividends, Interest Rates and Their Effect on Stock Options
Learn how analyzing dividends and interest rates is crucial to knowing when to exercise early. 
Trading
How to Build Valuation Models Like BlackScholes
Want to build a model like BlackScholes? Here are the tips and guidelines. 
Trading
Understanding Option Pricing
Before venturing into the world of trading options, investors should have a good understanding of the factors determining the value of an option. 
Trading
Factors That Determine Option Pricing
A thorough understanding of factors that affect price is essential in options trading. 
Trading
How and Why Interest Rates Affect Options
The Fed is expected to change interest rates soon. We explain how a change in interest rates impacts option valuations. 
Trading
Getting acquainted with options trading
Learn about trading stock options, including some basic options trading terminology.

How does implied volatility impact the pricing of options?
Learn about two specific volatility types associated with options and how implied volatility can impact the pricing of options. Read Answer >> 
How do I change my strike price once the trade has been placed already?
Learn how the strike prices for call and put options work, and understand how different types of options can be exercised ... Read Answer >> 
How is the stock market affected by Thanksgiving and Black Friday?
Thanksgiving and Black Friday sales numbers are considered to be important indicators for stock market activity throughout ... Read Answer >> 
What Happens to Call Options If a Co. is Bought?
Typically, the announcement of a buyout offer by another company is a good thing for shareholders. Read Answer >> 
How Do Speculators Profit From Options?
Options are a risky game, but you can learn speculators' tricks to use them to your advantage. Read Answer >>